Magnetic moment of an electron (NLO): ab-initio derivation

This section derives, from the QED Lagrangian and with no steps assumed, every integral that the NLO section (the next section, “Magnetic moment of an electron (NLO)”) evaluates. Nothing is computed here: the end products are the exact loop/parameter integrals — the LO massive-photon kernel, the five two-loop diagrams I, IIa, IIc, IId, IIe with their subtractions, and the projector that turns each of them into a number — together with a proof that their sum is the physical $A_2$ and an accounting of every other term the expansion generates (they all cancel, vanish, or renormalize parameters).

The logical chain, each link derived below:

1. QED Lagrangian with regulators (photon mass $\lambda$, Pauli–Villars) → free propagators and the interaction vertex.

2. The quantity: $a_e = F_2(0)$, with the form-factor decomposition and the explicit Breit-frame projector proved.

3. The 3-point Green function and the LSZ reduction with interacting fields: residue factors $Z_2$, amputation by full propagators, why external-leg diagrams (IIb/IIf) drop out.

4. Gell-Mann–Low formula (proved in the QFT section) + Dyson series + Wick’s theorem (proved here, with the fermion sign rules) → the completely systematic algebraic expansion.

5. The expansion executed at orders $e^1$, $e^3$, $e^5$: every single contraction enumerated and classified (mechanically verified by code/wick_enum.py).

6. The loop-integration toolkit derived: Feynman parameters, the shift, Wick rotation, the master integral table of code/loops.py, angular averages, Pauli–Villars.

7. The one-loop building blocks derived: the massive-photon vertex (kernel $K$), the self-energy $\Sigma$, the vacuum polarization $\hat\Pi$ and its spectral function $\rho(t)$, the one-loop vertex parts $\Lambda^\nu$ with their Feynman-parameter denominators.

8. The five raw two-loop integrals written out explicitly.

9. Renormalization: the on-shell counterterms, the Ward identity $Z_1 = Z_2$, and the assembly theorem: the Karplus–Kroll-subtracted diagrams of the NLO section sum exactly to the ab-initio $a_e\big|_{\alpha^2}$.

A note on terminology, since the question arises naturally: the “systematic algebraic expansion” has two distinct steps. Expanding the time-ordered exponential $S = T\exp(i\int\mathcal{L}_1)$ in powers of $e$ is the Dyson series (an ordinary Taylor expansion under the $T$-symbol). Converting each resulting vacuum expectation value of a $T$-product of free fields into a sum of products of propagators is Wick’s theorem. Feynman diagrams are nothing but the bookkeeping of the second step: one diagram per class of complete contractions. Both steps are carried out explicitly below.

1. The theory¶

We work in the conventions of the QED and LO sections: metric $(+,-,-,-)$, $\hbar = c = 1$, Dirac representation for $\gamma^\mu$, electron charge $e$ (only $e^2 = 4\pi\alpha$ ever appears). The Lagrangian density, including the two regulators used by Karplus–Kroll/Petermann and by all our computations, is

• The last term is the interaction $\mathcal{L}_1 = -e\bar\psi\slashed{A}\psi$ that follows from gauge coupling (QED section); in the Dyson series each factor $i\mathcal{L}_1$ produces the vertex factor $-ie\gamma^\mu$. Strictly the Lagrangian coupling is the bare charge $e_0$; the distinction ($e_0^2 = e^2\left[1-\Pi(0)\right]$, §3.3) matters only in the charge-renormalization bookkeeping of §3.3/§9, so we write $e$ throughout and reinstate $e_0$ exactly there.

• $\lambda$ is the photon mass, the infrared regulator. Together with the Feynman gauge-fixing term $-\frac12(\partial A)^2$ it gives the photon propagator its simple $-ig_{\mu\nu}$ numerator (derived below). This defines the IR regularization scheme of the 1950s papers: photon propagator $-ig_{\mu\nu}/(k^2-\lambda^2)$ everywhere, $\lambda \to 0$ at the end. (A genuine Proca photon would carry an extra $k_\mu k_\nu/\lambda^2$ term; by the collapse lemma of §2.4 such longitudinal pieces only produce $\gamma^\mu$-structures, i.e. they change $F_1$-type contact terms, and the $\lambda\to 0$ limit of the total $F_2$ is prescription independent. All per-diagram values in the NLO section refer to the $-ig_{\mu\nu}/(k^2-\lambda^2)$ prescription.)

• $m_0 = m - \delta m$ is the bare mass. We expand around the physical mass $m$:

  $$\mathcal{L} \supset -m_0\bar\psi\psi = -m\bar\psi\psi + \delta m\,\bar\psi\psi,$$

  (2)

  so propagators carry $m$ and the counterterm generates an extra two-point vertex $+i\,\delta m$ (one insertion is order $\alpha$, exactly what is needed at two loops). The on-shell condition fixing $\delta m$ is derived in §4.2.

• Ultraviolet regularization is Pauli–Villars, implemented at the level of the single logarithmically divergent master integral (§6.5): every log divergence is carried as the explicit symbol $L_{UV} = \log\Lambda^2 + \text{const}$, and every $F_2$ below must be (and is, assertedly) free of $L_{UV}$. For the fermion loop of the vacuum polarization the PV subtraction must be gauge invariant (two-mass PV); §7.3 explains where this matters and why the final answer is unambiguous.

1.1 Free propagators¶

The fermion propagator was derived in the QFT section:

For the photon we need the propagator that follows from our quadratic Lagrangian (mass term plus Feynman gauge fixing). Integrating by parts (dropping total derivatives, which do not change the action),

so the gauge-fixing term cancels the second piece exactly and

Each of the four components of $A_\mu$ is a free scalar of mass $\lambda$ (with a sign from the metric for the spatial ones), so the propagator is the inverse of the operator in brackets with the causal $i\epsilon$, i.e. in momentum space $-g^{\mu\nu}(k^2-\lambda^2)$ inverted times $i$:

This is the massless propagator of the QFT section with $k^2 \to k^2-\lambda^2$, as promised.

2. The quantity: $a_e = F_2(0)$¶

2.1 Setup¶

As in the LO section, the electron scatters off a fixed classical potential $A_\mu^{\mathrm{cl}}$; to first order in the potential and to all orders in the radiative corrections the amplitude is

and the anomalous moment is $a_e = F_2(0)$ where $F_2$ is defined by the decomposition of $\Gamma^\mu$ derived next. The connection between $F_{1,2}(0)$ and the $g$-factor ($g = 2(F_1+F_2)$, Born-level matching to $-\boldsymbol{\mu}\cdot\mathbf{B}$) was derived in full in the LO section and is not repeated.

2.2 The form-factor decomposition, derived¶

Claim. For on-shell spinors ($\slashed{p}u(p) = m\,u(p)$, $\bar u(p')\slashed{p}' = m\,\bar u(p')$, $p^2 = p'^2 = m^2$) and a parity-conserving theory, the most general vertex satisfying current conservation is

Step 1 — basis. $\Gamma^\mu$ is a $4\times4$ matrix carrying one Lorentz index, built from $\gamma^\mu$, $p$, $p'$, $g$, and (by parity conservation, no $\gamma^5$ or $\epsilon$-tensor) nothing else. Any product of $\slashed{p}$'s and $\slashed{p}'$'s sandwiched between the on-shell spinors collapses: commute $\slashed{p}$ to the right ($\slashed{p}\to m$) and $\slashed{p}'$ to the left ($\slashed{p}'\to m$) using $\slashed{a}\slashed{b} = 2a\cdot b - \slashed{b}\slashed{a}$; each commutation only produces scalars $p\cdot p' = m^2 - q^2/2$. The sandwiched vertex therefore reduces to

with scalar functions $A,B,C$ of $q^2$ (the only invariant). Structures $\sigma^{\mu\nu}q_\nu$ and $\sigma^{\mu\nu}P_\nu$ are not extra: writing $i\sigma^{\mu\nu}a_\nu = -\gamma^\mu\slashed{a}+a^\mu = \slashed{a}\gamma^\mu - a^\mu$ (direct consequence of $\sigma^{\mu\nu} = i(\gamma^\mu\gamma^\nu - g^{\mu\nu})$),

(The first line: $\bar u'(-\gamma^\mu\slashed{q}+q^\mu)u$ with $\slashed{q}u = (\slashed{p}'-\slashed{p})u = \slashed{p}'u - mu$ and $\gamma^\mu\slashed{p}' = 2p'^\mu - \slashed{p}'\gamma^\mu$, then $\bar u'\slashed{p}' = m\bar u'$; both identities are verified numerically below.)

Step 2 — current conservation. The Ward–Takahashi identity (derived ab initio in §4.3) gives $q_\mu\,\bar u'\Gamma^\mu u = 0$ on shell. Now $q\cdot P = p'^2 - p^2 = 0$ and $\bar u'\slashed{q}u = (m-m)\bar u'u = 0$ hold identically, so transversality forces $q^2\,C(q^2) = 0$, i.e. $C \equiv 0$.

Step 3 — repackaging. Use the Gordon identity to trade $P^\mu$ for $\gamma^\mu$ and $i\sigma^{\mu\nu}q_\nu$:

2.3 The explicit projector (as used by the pipeline)¶

All our computations extract $F_2$ by sandwiching the matrix $\Gamma^\mu$ between explicit spinors in the Breit frame (code/dirac.py),

Dirac-representation spinors ($\xi_0 = \binom10$, $\xi_1 = \binom01$, $\sigma^3\xi_s = \pm\xi_s$):

Define the two sandwiches used everywhere ($A_\mu$ here is a number, not a field):

Evaluate the two basis structures on these sandwiches. With $\mathbf{p}'\cdot\boldsymbol{\sigma} = -w\sigma^3$, $\mathbf{p}\cdot\boldsymbol{\sigma} = w\sigma^3$:

• $(\mu,s',s)=(0,0,0)$: $\bar u’\gamma^0 u = u’^\dagger u = (E+m),\xi_0^\dagger\xi_0

  • \frac{(-w\sigma^3)(w\sigma^3)}{E+m} = (E+m) - \frac{w^2}{E+m} = 2m$. For the$\sigma$-structure only$\nu=3$contributes ($q_3 = -q^3 = 2w$), and$\frac{i\sigma^{03}q_3}{2m} = \frac{i(i\gamma^0\gamma^3)(2w)}{2m} = -\frac{w}{m}\gamma^0\gamma^3$; using$u’^\dagger\gamma^3u = 2w$(same block algebra) the value is$-\frac{2w^2}{m}$.

• $(\mu,s',s)=(1,0,1)$: with $\gamma^0\gamma^1 = \begin{pmatrix}0&\sigma^1\\ \sigma^1&0\end{pmatrix}$ and $\sigma^1\xi_1 = \xi_0$: $\bar u'\gamma^1 u = u'^\dagger\gamma^0\gamma^1 u = -2w\,\xi_0^\dagger\sigma^1\xi_1 = -2w$; and $\frac{i\sigma^{13}q_3}{2m} = -\frac{w}{m}\gamma^1\gamma^3$ with $\gamma^1\gamma^3 = i\sigma^2\,\mathbb{1}_{2\times2\,\text{blocks}}$, $i\sigma^2\xi_1 = \xi_0$, gives $-\frac{w}{m}\left[(E+m)-\frac{w^2}{E+m}\right] = -2w$.

So on these two sandwiches

— the $\mu=1$ flip sandwich alone measures only the magnetic combination $F_1+F_2$, which is why it must be paired with the $\mu=0$ sandwich. Solving the $2\times2$ linear system:

exact at finite $w$. As $w\to0$ (i.e. $q^2\to0$), $A_1$ vanishes linearly in $w$ (it is proportional to $w(F_1+F_2)$), so with $m=1$

which is precisely the projection formula quoted and used in the NLO section (code/g2_iid.py validates it on the LO diagram). Everything above is verified symbolically:

>>> import sys; sys.path.insert(0, 'code')
>>> from sympy import symbols, simplify
>>> from dirac import formfactor_basis
>>> m, w = symbols("m w", positive=True)
>>> [simplify(b) for b in formfactor_basis(m, w, 0, 0, 0)]
[2*m, -2*w**2/m]
>>> [simplify(b) for b in formfactor_basis(m, w, 1, 0, 1)]
[-2*w, -2*w]

2.4 The collapse lemma (longitudinal photon pieces)¶

For later use (gauge/regulator independence, Ward manipulations): for any internal-photon momentum $k$ attached to the electron line next to an on-shell end,

Iterating such telescoping steps, a $k_\mu k_\nu$ numerator on an internal photon collapses the two adjacent propagator chains and leaves only $\gamma^\mu$-type (i.e. $F_1$-type) structures. Both identities are checked numerically in the pipeline conventions (they are two-line consequences of $(\slashed{p}-m)u(p)=0$).

3. Green functions, LSZ with interacting fields, and amputation¶

3.1 The 3-point function¶

Exactly as in the LO section, the amplitude is obtained from

with the external legs stripped by the reduction rules of the QFT section. Those rules were derived there with free-field residues; at two loops we need the refinement that the interacting field creates the physical one-electron state with amplitude

$Z_2$ is defined by this equation. Repeating the free-field computation of the propagator with this matrix element (insert a complete set of states into $\langle T\psi\bar\psi\rangle$; the one-particle contribution reproduces the free two-point integral times $Z_2$, with the physical mass $m$ in the exponentials; multi-particle states start at the cut $p^2 \ge (m+\lambda)^2$) gives the exact two-point function near its pole:

Consequently the correct reduction rule for each external fermion leg is the QFT-section rule divided by $\sqrt{Z_2}$ (the free rule assumed unit amplitude), with $m$ the physical mass.

3.2 Two-point functions from 1PI blocks¶

Let $-i\Sigma(\slashed{p})$ denote the sum of all one-particle- irreducible (1PI) self-energy insertions (plus the $+i\delta m$ counterterm vertex, so $\Sigma_{\rm eff} = \Sigma - \delta m$; we write $\Sigma$ for $\Sigma_{\rm eff}$ from here on). Summing the geometric series of blobs strung on free propagators,

Matching to the pole form above:

Identically for the photon: with $i\Pi^{\mu\nu}(q) = (q^2g^{\mu\nu}-q^\mu q^\nu)\,i\Pi(q^2)$ the 1PI photon self-energy (transversality is derived in §7.3), the dressed propagator sums to

(displayed at $\lambda=0$; the $q_\mu q_\nu$ terms never contribute by §2.4 and the Ward identity).

3.3 Factorized form of $\tilde G^\mu$ and the S-matrix vertex¶

Every connected diagram of $\tilde G^\mu$ decomposes uniquely as (blobs on the $p$ leg) ⊗ (blobs on the $p'$ leg) ⊗ (vacuum-polarization chain on the photon leg) ⊗ (core vertex 1PI in all three legs). This is not an assumption: at the orders we need it is verified term by term by the complete enumeration of §5 — every contraction lands in exactly one class (the odd-loop class V would be additional core diagrams, but it cancels pairwise by Furry’s theorem, §5). Therefore, exactly,

where $\Gamma^\nu$ is the proper (1PI) vertex — at tree level $\gamma^\nu$; at one loop the LO diagram; at two loops the five diagrams of §8.

Now apply the corrected reduction of §3.1. Each fermion leg contributes

(using $(\slashed{p}'-m)(\slashed{p}'+m) = p'^2-m^2$; the regular part of $\tilde S'$ is killed by $(\slashed{p}'-m)u = 0$), and similarly $\sqrt{Z_2}\,u(p)$ on the incoming side. The photon leg is the external potential; its propagator and $1/(1-\Pi)$ dressing combine with the charge as follows. On-shell charge renormalization defines

so that $e_0^2/(1-\Pi(q^2)) = e^2\left[1+\hat\Pi(q^2)+O(\alpha^2)\right]$. Collecting everything:

Three structural conclusions, each an exact statement about classes of diagrams:

1. External-leg self-energies (KK classes IIb, IIf and all their two-loop variants) never enter the vertex. They are part of $\tilde S'(p^{(\prime)})$ and are consumed by the reduction, which leaves only the number $Z_2$ per leg — total factor $Z_2$.

2. Vacuum-polarization chains on the external-potential line (KK classes III and IV) are the factor $1/(1-\Pi(q^2))$. After charge renormalization they contribute $F_i^{\rm blob}(q^2)\,\hat\Pi(q^2)$; since $\hat\Pi(0) = 0$ and all $F_i^{\rm blob}$ are finite at $q^2 = 0$, they contribute nothing to $a_e = F_2(0)$: pure charge renormalization.

3. The physical form factors are those of $Z_2\,\Gamma^\mu$:

   $$a_e = F_2^{S}(0) = Z_2\;F_2(0),\qquad F_1^{S}(0) = Z_2\,F_1(0).$$

   (33)

Expanding to order $\alpha^2$ with $F_2(0) = F_2^{(2)}(0) + F_2^{(4)}(0) + \dots$ (superscripts count powers of $e$) and $Z_2 = 1+\delta Z_2$:

This is the master ab-initio formula. By the Ward identity (§4.3) $\delta Z_2 = -\delta F_1(0)$, so the second term is exactly the “vertex renormalization cross term $-\delta F_1(0)\cdot\frac{\alpha}{2\pi}$” of the NLO section.

4. The expansion machinery¶

4.1 Gell-Mann–Low and the Dyson series¶

From the QFT section (proved there),

with here $\mathcal{L}_1 = -e\bar\psi\slashed{A}\psi

• \delta m,\bar\psi\psi$(all fields interaction-picture free fields). Taylor-expanding the exponential under the$T$symbol (legitimate because everything inside one$T$-product commutes by construction) gives the order-$e^n$ term of the 3-point function:

plus, at each order in $\alpha$, the terms with $\delta m$ insertions. Only odd $n$ survives (the photon fields must pair up: $n$ vertex photons + 1 external photon must be even). The vacuum diagrams generated simultaneously in $\langle0|S|0\rangle$ cancel: any complete contraction factorizes uniquely into its externals-connected part (say $n_1$ vertices) and its vacuum part ($n_2 = n-n_1$ vertices); summing over which labeled vertices go where turns $1/n!\binom{n}{n_1}$ into $\frac{1}{n_1!}\frac{1}{n_2!}$, so the numerator factorizes as (no-vacuum series) × $\langle0|S|0\rangle$ and the denominator removes the second factor exactly.

4.2 Wick’s theorem (with fermion signs)¶

Definitions. For a free field $\phi = \phi^+ + \phi^-$ ($\phi^+$ annihilates, $\phi^-$ creates), the normal product $N\{\cdots\}$ puts all $\phi^-$ left of all $\phi^+$, with a factor $(-1)$ for each transposition of two fermionic fields performed in the rearrangement. The contraction of two fields is the c-number

Because $[\phi^+(x),\phi^-(y)]_{\mp}$ is a c-number for free fields, the contraction is a c-number, and taking $\langle0|\cdot|0\rangle$ (normal products of one or more fields have zero vacuum expectation value) shows it equals the propagator: $\underbrace{\psi_a(x)\bar\psi_b(y)} = S_{ab}(x-y)$, $\underbrace{A_\mu(x)A_\nu(y)} = D_{\mu\nu}(x-y)$, and $\underbrace{\psi\psi} = \underbrace{\bar\psi\bar\psi} = 0$.

Theorem (Wick).

where the sign of each term is the parity of the permutation that brings every contracted pair into adjacent order (fermions anticommute, bosons commute, in this counting).

Proof (induction in $n$). For $n=2$ this is the definition of the contraction. Assume the statement for $n-1$ fields. Both sides change by the same graded sign under any permutation of the fields (that is how $T$, $N$, and the pair-adjacency sign are all defined), so we may relabel such that $t_1$ is the earliest time; then $T\{\phi_1\cdots\phi_n\} = T\{\phi_2\cdots\phi_n\}\,\phi_1$, and by the induction hypothesis the right-hand side is [the signed contraction sum of $\phi_2\cdots\phi_n$ inside normal products] $\times\,\phi_1$. Split $\phi_1 = \phi_1^+ + \phi_1^-$. The annihilation part $\phi_1^+$ already stands in normal position at the far right. The creation part $\phi_1^-$ must be moved to the far left; each transposition past the annihilation part $\phi_k^+$ of an uncontracted field produces the graded commutator $[\phi_k^+,\phi_1^-]_\mp$ — a c-number equal to $\langle0|\phi_k\phi_1|0\rangle = \langle0|T\phi_k\phi_1|0\rangle$ (since $t_k \ge t_1$), i.e. exactly the contraction $\underbrace{\phi_k\phi_1}$ — while the transposition signs accumulate into precisely the pair-adjacency sign of that new contraction. Collecting the terms with and without the new contraction reproduces the claimed sum for $n$ fields. $\;\blacksquare$

Taking the vacuum expectation value kills every term with uncontracted fields:

Sign rules (consequences used constantly below):

• Open fermion line. In the canonical operator order $\psi_{\rm out}\,\bar\psi_{\rm in}\, (\bar\psi_1\psi_1)(\bar\psi_2\psi_2)\cdots$ every contraction whose fermion pairs form only an open line from $\bar\psi_{\rm in}$ to $\psi_{\rm out}$ has total sign +1.

• Closed fermion loop. Each closed loop contributes one factor $(-1)$ and a Dirac trace. Derivation on the shortest case, the two-vertex loop $\langle(\bar\psi\Gamma^{(1)}\psi)_x(\bar\psi\Gamma^{(2)}\psi)_y\rangle$: the only contraction is $\bar\psi_a(x)\Gamma^{(1)}_{ab}\psi_b(x)\, \bar\psi_c(y)\Gamma^{(2)}_{cd}\psi_d(y)$ with $\underbrace{\psi_b(x)\bar\psi_c(y)} = S_{bc}(x-y)$ and the pair $(\bar\psi_a,\ \psi_d)$ contracted in reversed order: bringing $\psi_d$ in front of $\bar\psi_a$ costs the T-reordering sign $\langle T\bar\psi_a\psi_d\rangle = -S_{da}(y-x)$, so the term equals $-\,S_{da}(y-x)\Gamma^{(1)}_{ab}S_{bc}(x-y)\Gamma^{(2)}_{cd} = -\,\mathrm{tr}\left[\Gamma^{(1)}S(x-y)\Gamma^{(2)}S(y-x)\right]$. The general loop works identically (one reversed pair per loop).

Both rules — and nothing else about signs — are verified for every single contraction through two loops by code/wick_enum.py (§5), which computes each term’s permutation sign explicitly and asserts $\text{sign} = (-1)^{\#\text{closed loops}}$.

4.3 The Ward–Takahashi identity, $Z_1 = Z_2$¶

The fermion equation of motion from §1, $(i\slashed{\partial} - m_0 - e\slashed{A})\psi = 0$ and its conjugate, give current conservation as an operator equation regardless of the photon mass (the $\slashed{A}$ and mass terms cancel between the pair):

Differentiate the time-ordered product (the $T$-symbol’s $\theta$-functions produce equal-time commutators):

The first term vanishes; the equal-time commutators follow from $\{\psi_a(\mathbf{z}),\psi_b^\dagger(\mathbf{x})\} = \delta_{ab}\delta^{(3)}(\mathbf{z}-\mathbf{x})$ via $[AB,C] = A\{B,C\} - \{A,C\}B$:

Fourier transforming (conventions of the QFT section) and amputating the two fermion legs with full propagators turns this into the momentum-space Ward–Takahashi identity for the proper vertex; the overall normalization is fixed by the tree-level check ($\Gamma^\mu = \gamma^\mu$, $\tilde S'^{-1} = -i(\slashed{p}-m)$):

Two consequences used above and below:

• On shell both sides sandwiched with $\bar u(p')\dots u(p)$ vanish: $q_\mu\bar u'\Gamma^\mu u = 0$ (used in §2.2).

• $q\to0$: write $\Sigma(\slashed{p}) = A(p^2)+B(p^2)\slashed{p}$ and expand the identity to first order in $q$; sandwiching and using the $q=0$ Gordon identity $\bar u\, p^\mu\, u = m\,\bar u\gamma^\mu u$ to convert the $P^\mu$-structures,

  $$F_1(0) = 1 - \left[B + 2mA' + 2m^2B'\right]_{p^2=m^2} = 1 - \Sigma'(m) = 1 - \delta Z_2 .$$

  (44)

  Hence with $F_1(0) \equiv 1 + \delta F_1(0)$ and $Z_1^{-1} \equiv F_1(0)$:

  $$\boxed{\ \delta F_1(0) = -\,\delta Z_2,\qquad Z_1 = Z_2 + O(\alpha^2).\ }$$

  (45)

  (The identity holds to all orders; we only need it at order $\alpha$.) In particular $F_1^S(0) = Z_2F_1(0) = 1$ exactly: the charge is not renormalized by the vertex, and the LO section’s renormalization condition “$F_1(0)=1$ to all orders” is automatic.

4.4 Momentum-space Feynman rules (assembled once)¶

Applying Wick’s theorem to the Dyson term of §4.1 and Fourier transforming every propagator, each $\int\mathrm{d}^4x_i$ produces $(2\pi)^4\delta^{(4)}(\sum k_{\rm in})$ at vertex $i$; one delta survives as overall momentum conservation and the remaining momentum integrals are the loop integrals. The dictionary (all derived above):